Minimal triangulations for an infinite family of lens spaces

TL;DR

This paper proves that for an infinite family of lens spaces, specifically L(2n,1), the minimal layered triangulation is unique and also the minimal triangulation, determining their complexity.

Contribution

It establishes the uniqueness and minimality of layered triangulations for all lens spaces L(2n,1), extending understanding of their triangulation complexity.

Findings

Minimal layered triangulation is unique for L(2n,1) lens spaces.

Minimal triangulations are determined for an infinite family of lens spaces.

The complexity of these lens spaces is explicitly characterized.

Abstract

The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its "minimal layered triangulation." This paper proves that for each integer n>1, the minimal layered triangulation of the lens space L(2n,1) is its unique minimal triangulation. More generally, the minimal triangulations (and hence the complexity) are determined for an infinite family of lens spaces containing the lens spaces L(2n,1).